Hydrodynamic Analysis and Optimization of a Hinged-type Wave Energy Converter -
SeaWEED
by c Meng Chen
A thesis submitted to the School of Graduate Studies
in partial fulfilment of the requirements for the degree of
Master of Engineering
Department of Engineering and Applied Science Memorial University of Newfoundland
May 2020
St. John’s Newfoundland
Abstract
This thesis presents the experimental, numerical and optimization studies on a hinged-type wave energy converter, SeaWEED (Sea Wave Energy Extraction Device), developed by Grey Island Energy. The device is considered as an improved attenuator consisting of four modules connected by adjustable truss structures.
Extensive model tests of a 1:35 scale SeaWEED model with and without the power-take-off (PTO) units have been conducted at the towing tank of Memorial Uni- versity (MUN). Friction dampers were designed to mimic the PTO systems. Repeated tests were carried out at a few wave frequencies around the region with maximum responses, and good repeatability has been observed.
Potential-flow based time- and frequency- domain programs utilizing the Lagrange multiplier approach have been developed to simulate the dynamics of SeaWEED. In the time-domain program, nonlinear Froude-Krylov forces are calculated over the instantaneous wetted surfaces of the bodies under the wave profile, and the Wheeler Stretching method is applied to compute the wave pressure. The numerical results are compared with the experimental data, and good agreement is achieved.
Optimization studies have been further conducted utilizing the frequency-domain
program. Various parameters, including damping coefficients of the PTO systems,
lengths of truss structures and the draft of the device, are considered. The uniform
design method is used for sampling, and the response surface method is employed for
surrogate construction. The desirability optimization method is utilized to optimize
the response. An optimal combination of parameters is determined for an intended
operation site.
Acknowledgements
First of all, I would like to express my deepest gratitude to my supervisor, Dr.
Heather Peng, for her patient and invaluable academic guidance, enthusiastic encour- agement, and kindness throughout my study at MUN. With her insight, support, and suggestions, I have developed a good knowledge and great interest in my research field.
I am also grateful that I have the opportunity to involve in the SeaWEED project.
I would also like to thank Dr. Wei Qiu, for his professional advice and support for this work. Taking his two insightful courses helped me develop my programming skills and professional knowledge in marine hydrodynamics.
My gratitude is also extended to Brian Lundrigan, Trevor Clark, Matt Curtis, and Craig Mitchell for their kind help during the model tests. I would also like to thank the work term students, Min Zhang, Kyle Stanley, Craig Thompson, Sara White, Jackie Zhang, Florence Panisset, and Simon Xu, for their contribution to the project.
Furthermore, I am grateful to my colleagues in AMHL, especially Yuzhu Li, who helped to me at the early stages of the SeaWEED project.
I would like to give special thanks to Wei Meng, who is my colleague and also my
husband, for his support and concern on my academic research and my life. Moreover,
I would like to thank my parents and parents-in-law for their unconditional love,
support, and understanding during my entire study.
Contents
Abstract ii
Acknowledgements iii
List of Tables viii
List of Figures ix
1 Introduction 1
1.1 Background . . . . 1
1.1.1 Wave Energy Converters . . . . 2
1.1.1.1 Oscillating Water Columns . . . . 3
1.1.1.2 Oscillating Bodies . . . . 5
1.1.1.3 Overtopping Devices . . . . 7
1.1.2 SeaWEED (Sea Wave Energy Extraction Device) . . . . 9
1.1.3 Constrained Dynamics . . . . 9
1.1.4 Optimization of Simulation-based Design . . . . 11
1.1.4.1 Selecting Sampling Points . . . . 12
1.1.4.2 Constructing and Exploiting a Surrogate . . . . 14
1.1.4.3 Wave Energy Converter Optimization . . . . 16
1.2 Overview . . . . 17
2 Introduction and Model Tests of SeaWEED 19 2.1 Introduction of SeaWEED . . . . 19
2.2 Model Tests on the First Generation of SeaWEED . . . . 22
2.3 Model Tests on the Second Generation of SeaWEED . . . . 25
2.3.1 Test Matrix . . . . 27
2.3.2 Test Set-up . . . . 30
2.3.3 Pre-tests . . . . 31
2.3.3.1 Wave Probe Calibration . . . . 32
2.3.3.2 Load Cell Calibration . . . . 32
2.3.3.3 Displacement Sensor Calibration . . . . 35
2.3.3.4 Damper Calibration . . . . 35
2.3.3.5 Sychronization of Measurements . . . . 37
2.3.4 Experimental Data-processing . . . . 39
3 Time-domain Simulation of SeaWEED 43 3.1 Mathematical Formulation . . . . 44
3.1.1 Coordinate System Definition . . . . 44
3.1.2 Equations of Motion . . . . 44
3.1.3 Computation of Constraint Forces . . . . 47
3.1.4 Computation of Nonlinear Froude-Krylov Forces . . . . 50
3.1.5 Convergence Studies . . . . 52
3.1.5.1 Convergence Studies on Mesh . . . . 52
3.1.5.2 Convergence Studies on Time Step . . . . 53
3.2 Validation Studies . . . . 54
3.2.1 Free Hinge Conditions . . . . 55
3.2.2 Damped Conditions . . . . 60
4 Frequency-domain Simulation of SeaWEED 72 4.1 Mathematical Formulation . . . . 72
4.1.1 Equations of Motion and Constraint Matrix . . . . 73
4.1.2 Power Absorption in Regular and Irregular Waves . . . . 75
4.2 Validation Studies . . . . 77
4.2.1 First Generation SeaWEED Simulation . . . . 77
4.2.2 Second Generation SeaWEED Simulation . . . . 79
4.2.2.1 Free-hinged Conditions . . . . 80
4.2.2.2 Damped Conditions . . . . 84
5 Optimization of the Second Generation of SeaWEED 95 5.1 Optimization Problem . . . . 96
5.2 Selection of Sample Points . . . . 96
5.3 Surrogate Model Construction . . . . 97
5.4 Optimal Response Exploration . . . . 98
5.5 Optimization of SeaWEED . . . 100
5.5.1 Optimization Problem Definition . . . 100
5.5.2 Selection of Sampling Points and Levels . . . 102
5.5.3 Optimization Results . . . 106
6 Conclusions and Future Work 110
Bibliography 112
List of Tables
2.1 Particulars of SeaWEED (first generation) . . . . 24
2.2 Particulars of SeaWEED (second generation) . . . . 27
2.3 Truss length combinations . . . . 28
2.5 Damping cases . . . . 28
2.4 Test matrix . . . . 29
2.6 Scaling of quantities . . . . 42
5.1 Geometrical design variables (full-scale) . . . 101
5.2 PTO damping coefficients (full-scale) . . . 101
5.3 Geometrical design variable level combination . . . 103
5.4 CD value . . . 103
5.5 Adjusted R-squared value . . . 104
5.6 Average error . . . 106
5.7 Optimal parameters (full-scale) . . . 107
List of Figures
1.1 Category of WECs Based on Working Principles (IEA, 2012) . . . . . 2
1.2 WEC Working Principles (Day et al., 2015) . . . . 3
1.3 Fixed OWCs (Falc˜ao et al., 2000 and Torre-Enciso et al., 2009) . . . . 4
1.4 LeanCon WEC (LeanCon) . . . . 4
1.5 Pelamis (Yemm et al., 2012) . . . . 5
1.6 Point Absorbers . . . . 6
1.7 Oscillating Wave Surge Converters (AW-Energy and Renzi et al., 2014) 7 1.8 Seawave Slot-Cone Generator (Vicinanza et al., 2008) . . . . 8
1.9 Wave Dragon (Kofoed et al., 2006) . . . . 8
1.10 The First Generation of SeaWEED . . . . 10
1.11 The Second Generation of SeaWEED . . . . 10
2.1 SeaWEED PTO System . . . . 21
2.2 SeaWEED Ballast Tank . . . . 22
2.3 General Arrangement of the Basin (NRC) . . . . 23
2.4 SeaWEED Model in the NRC-IOT Basin . . . . 23
2.5 Body and Tie-rod Length Definition . . . . 24
2.6 Towing Tank of MUN . . . . 26
2.7 SeaWEED Model in the Towing Tank . . . . 26
2.8 Body and Truss Length Definition . . . . 27
2.9 Test Set-up . . . . 30
2.10 Friction Damper . . . . 31
2.11 Friction Damper . . . . 32
2.12 Wave Probe Calibration . . . . 33
2.13 Wave Probe Calibration Results (WP3) . . . . 33
2.14 Honeywell Model-31 Load Cell . . . . 34
2.15 Load Cell Calibration Results (PTO-2) . . . . 34
2.16 Displacement Sensor Calibration Results (PTO-2) . . . . 35
2.17 Damper Calibration Apparatus . . . . 36
2.18 Damper Calibration Results (D1) . . . . 37
2.19 Damper Calibration Results (D2, D3 and D4) . . . . 38
2.20 Damper Calibration Results (D5) . . . . 38
2.21 Synchronization Input by Module NI9264 . . . . 39
2.22 Relative Pitch Angles . . . . 40
2.23 Damping Force and Sliding Velocity . . . . 40
3.1 Coordinate Systems . . . . 44
3.2 Damping Moments . . . . 47
3.3 SeaWEED Wetted Surfaces under Waves . . . . 50
3.4 Illustration of Wheeler Stretching Theory . . . . 51
3.5 Convergence Study on the Mesh Size . . . . 53
3.6 Time Histories of Simulated Hinge Motions . . . . 54
3.7 Time Histories of Relative Displacement Error . . . . 55
3.8 Non-dimensional Relative Pitch Angles at PTO-1 (Case T1-D0) . . . 56
3.9 Non-dimensional Relative Pitch Angles at PTO-2 (Case T1-D0) . . . 57
3.10 Non-dimensional Relative Pitch Angles at PTO-1 (Case T2-D0) . . . 57
3.11 Non-dimensional Relative Pitch Angles at PTO-2 (Case T2-D0) . . . 58
3.12 Non-dimensional Relative Pitch Angles at PTO-1 (Case T3-D0) . . . 58
3.13 Non-dimensional Relative Pitch Angles at PTO-2 (Case T3-D0) . . . 59
3.14 Non-dimensional Relative Pitch Angles at PTO-1 (Case T4-D0) . . . 59
3.15 Non-dimensional Relative Pitch Angles at PTO-2 (Case T4-D0) . . . 60
3.16 Non-dimensional Relative Pitch Angles at PTO-1 (Case T1-D1) . . . 61
3.17 Non-dimensional Relative Pitch Angles at PTO-2 (Case T1-D1) . . . 62
3.18 Full-scale Capture Width at PTO-1 (Case T1-D1) . . . . 62
3.19 Full-scale Capture Width at PTO-2 (Case T1-D1) . . . . 63
3.20 Non-dimensional Relative Pitch Angles at PTO-1 (Case T1-D2) . . . 63
3.21 Non-dimensional Relative Pitch Angles at PTO-2 (Case T1-D2) . . . 64
3.22 Full-scale Capture Width at PTO-1 (Case T1-D2) . . . . 64
3.23 Full-scale Capture Width at PTO-2 (Case T1-D2) . . . . 65
3.24 Non-dimensional Relative Pitch Angles at PTO-1 (Case T1-D3) . . . 65
3.25 Non-dimensional Relative Pitch Angles at PTO-2 (Case T1-D3) . . . 66
3.26 Full-scale Capture Width at PTO-1 (Case T1-D3) . . . . 66
3.27 Non-dimensional Relative Pitch Angles at PTO-1 (Case T1-D4) . . . 67
3.28 Non-dimensional Relative Pitch Angles at PTO-2 (Case T1-D4) . . . 68
3.29 Full-scale Capture Width at PTO-2 (Case T1-D4) . . . . 68
3.30 Non-dimensional Relative Pitch Angles at PTO-1 (Case T1-D5) . . . 69
3.31 Non-dimensional Relative Pitch Angles at PTO-2 (Case T1-D5) . . . 69
3.32 Full-scale Capture Width at PTO-1 (Case T1-D5) . . . . 70
3.33 Full-scale Capture Width at PTO-2 (Case T1-D5) . . . . 70
4.1 Relative Pitch Angles . . . . 74
4.2 Non-dimensional Relative Pitch Angles at PTO-1 . . . . 78
4.3 Non-dimensional Relative Pitch Angles at PTO-1 . . . . 78
4.4 Full-scale Capture Width at PTO-1 . . . . 79
4.5 Non-dimensional Relative Pitch Angles at PTO-1 (Case T1-D0) . . . 80
4.6 Non-dimensional Relative Pitch Angles at PTO-2 (Case T1-D0) . . . 81
4.7 Non-dimensional Relative Pitch Angles at PTO-1 (Case T2-D0) . . . 81
4.8 Non-dimensional Relative Pitch Angles at PTO-2 (Case T2-D0) . . . 82
4.9 Non-dimensional Relative Pitch Angles at PTO-1 (Case T3-D0) . . . 82
4.10 Non-dimensional Relative Pitch Angles at PTO-2 (Case T3-D0) . . . 83
4.11 Non-dimensional Relative Pitch Angles at PTO-1 (Case T4-D0) . . . 83
4.12 Non-dimensional Relative Pitch Angles at PTO-2 (Case T4-D0) . . . 84
4.13 Non-dimensional Relative Pitch Angles at PTO-1 (Case T1-D1) . . . 85
4.14 Non-dimensional Relative Pitch Angles at PTO-2 (Case T1-D1) . . . 85
4.15 Full-scale Capture Width at PTO-1 (Case T1-D1) . . . . 86
4.16 Full-scale Capture Width at PTO-2 (Case T1-D1) . . . . 86
4.17 Non-dimensional Relative Pitch Angles at PTO-1 (Case T1-D2) . . . 87
4.18 Non-dimensional Relative Pitch Angles at PTO-2 (Case T1-D2) . . . 87
4.19 Full-scale Capture Width at PTO-1 (Case T1-D2) . . . . 88
4.20 Full-scale Capture Width at PTO-2 (Case T1-D2) . . . . 88
4.21 Non-dimensional Relative Pitch Angles at PTO-1 (Case T1-D3) . . . 89
4.22 Non-dimensional Relative Pitch Angles at PTO-2 (Case T1-D3) . . . 90
4.23 Full-scale Capture Width at PTO-1 (Case T1-D3) . . . . 90
4.24 Non-dimensional Relative Pitch Angles at PTO-1 (Case T1-D4) . . . 91
4.25 Non-dimensional Relative Pitch Angles at PTO-2 (Case T1-D4) . . . 91
4.26 Full-scale Capture Width at PTO-2 (Case T1-D4) . . . . 92
4.27 Non-dimensional Relative Pitch Angles at PTO-1 (Case T1-D5) . . . 92
4.28 Non-dimensional Relative Pitch Angles at PTO-2 (Case T1-D5) . . . 93
4.29 Full-scale Capture Width at PTO-1 (Case T1-D5) . . . . 93
4.30 Full-scale Capture Width at PTO-2 (Case T1-D5) . . . . 94
5.1 Wave Climate Table off Cork Harbour in Ireland . . . 100
5.2 Damping Optimization . . . 102
5.3 Verification of RSM . . . 105
5.4 Response Surfaces for Annual Power Absorption . . . 108
5.5 Power Matrix of SeaWEED . . . 109
Chapter 1
Introduction
1.1 Background
During the past decades, renewable wave energy has become a research focus due
to the ever-increasing energy consumption demand and environmental issues. Waves
are created by wind blowing over the ocean surface and can travel a great distance
with little energy loss (Cl´ement et al., 2002). Comparing to other forms of renewable
energies, wave energy has a relatively less adverse impact on the environment, larger
energy density and power output efficiency (Drew et al., 2009). The globally available
wave power is estimated as 3.7 TW, which is in the same order of magnitude of
the world consumption of electrical energy (Mørk et al., 2010). Although wave-
induced electricity generation is not well-developed currently (Pecher et al., 2017), its
outstanding benefits encourage many countries to devote efforts into the wave energy
conversion field.
1.1.1 Wave Energy Converters
Wave energy converters (WECs) are devices that can convert wave energy and generate electricity or other forms of energies (Day et al., 2015). Girard and son proposed the earliest patent of WEC (1799) and the first experiment of WEC was conducted during the 1800s (Gonzalez). In the 1970s, the remarkable energy crisis and a paper by Salter (1974) roused the research interest on wave power at sea. Since then, more than 1000 WEC patents have been proposed (Drew et al., 2009).
WECs can be categorized according to their working principles: oscillating wa- ter columns (OWCs), oscillating bodies (OBs), and overtopping devices (Day et al., 2015). Representative WECs of each categorization are given in Fig. 1.1, and the working principles of them are given in Fig. 1.2.
Figure 1.1: Category of WECs Based on Working Principles (IEA, 2012)
Figure 1.2: WEC Working Principles (Day et al., 2015)
1.1.1.1 Oscillating Water Columns
An OWC WEC consists of a semi-submerged chamber open to the sea below. As waves oscillate in the chamber, the air is forced out of the chamber and back into it.
The high-velocity air then drives the turbine to generate power, as shown in Fig. 1.2.
OWC devices can be fixed on the seabed, which makes them convenient and economical to construct and maintain, such as the Pico power plant (Falc˜ao et al., 2000) and the Mutriku breakwater wave plant (Torre-Enciso et al., 2009), as shown in Fig. 1.3.
On the other hand, The LeanCon WEC (Kofoed et al., 2008) is a large floating
type OWC, which consists of a large number of chambers, as shown in Fig. 1.4. The
device covers more than one wave length and utilizes an ordinary wind turbine to
generate power. In addition, due to the unique design of non-return valves, the air
flow can be rectified before the flow reaches the generator (Kofoed et al., 2008).
Figure 1.3: Fixed OWCs (Falc˜ao et al., 2000 and Torre-Enciso et al., 2009)
Figure 1.4: LeanCon WEC (LeanCon)
1.1.1.2 Oscillating Bodies
Oscillating body (OB) WECs harvest energy from the relative motions between bodies driven by waves. OB WECs have diverse forms, and they can be further categorized into attenuators, point absorbers, and oscillating wave surge converters, as shown in Fig. 1.2.
The Pelamis WEC, as shown in Fig. 1.5, is a semi-submerged floating attenuator, which converts wave energy into electricity. Its articulated structure is made of five tube sections by the universal hinge joints (Yemm et al., 2012). The device weather- vanes in head seas, and the relative pitch motions drive the hydraulic power-take-off (PTO) systems at the hinge joints.
Figure 1.5: Pelamis (Yemm et al., 2012)
Examples of point absorbers include PowerBuoy (Edwards et al., 2014), Seabased
(Seabased Technology) and Wavebob (Mouwen, 2008), as shown in Fig. 1.6. Power-
Buoy is a floating WEC which consists of a heave plate rigidly connected to a spar
and a float moving along the spar. The float oscillates in response to waves, and the
relative motions between the float and the spar drive the PTO hydraulic system and
generate power. Wavebob consists of two oscillating structures, a torus and a tank, and power is generated from the relative motions between the two parts. Different from PowerBuoy and Wavebob which float in the sea, Seabased has a base fixed on the seabed.
Figure 1.6: Point Absorbers
Among oscillating wave surge converters, examples are WaveRoller (AW-Energy)
and Oyster (Renzi et al., 2014), as shown in Fig. 1.7. The lower parts of devices are
anchored on the sea bed or fixed to a submerged floating reference frame; while the
upper parts move back and forth due to wave surge to drive hydraulic piston pumps
to generate energy.
Figure 1.7: Oscillating Wave Surge Converters (AW-Energy and Renzi et al., 2014)
1.1.1.3 Overtopping Devices
Overtopping devices capture power as waves flow up a ramp and over the top into a storage reservoir and the water passes through turbines, as shown in Fig. 1.2. Typical examples include the fixed-type Seawave Slot-Cone Generator (SSG) (Margheritini et al., 2009) and floating-type Wave Dragon (Kofoed et al., 2006).
The SSG device is designed to be fixed offshore, as presented in Fig. 1.8, and it is equipped with three reservoirs with different heights to increase power generation efficiency.
As shown in Fig. 1.9, the Wave Dragon WEC has a curved ramp, a large floating
reservoir, a platform with two reflectors for concentrating the power of incoming
waves, and several low-head hydro turbines (Kofoed et al., 2006).
Figure 1.8: Seawave Slot-Cone Generator (Vicinanza et al., 2008)
Figure 1.9: Wave Dragon (Kofoed et al., 2006)
1.1.2 SeaWEED (Sea Wave Energy Extraction Device)
SeaWEED is an attenuator-type WEC proposed by Grey Island Energy Inc.
(GIE), which consists of four modules that are connected by rigid truss structures.
The four-module array includes a non-energy producing nose module in the front, fol- lowed by two energy producing modules, and another non-energy producing module at the rear.
Initial conceptual studies have been carried out to evaluate the performance of the first generation device (see Fig. 1.10) by testing a 1:16 scale model in the wave basin of National Research Council Canada. The experimental and numerical results for the first generation model led to the second generation (see Fig. 1.11) with improvements in the hull geometry, a lower draft, and a different connection structure.
The device is considered as an improved attenuator in comparison with Pelamis.
The use of the rigid truss structure would allow for a higher power output per unit mass, and also reduce the side loading due to tidal currents, local wind or bi-modal swells in comparison with other attenuator devices. Additionally, the trusses can be customized in length to archive high efficiency for a specified site. The design also attempted to address the wave topping and slamming issues encountered by devices such as Pelamis.
1.1.3 Constrained Dynamics
SeaWEED undergoes constrained motions in waves due to the hinged joints. It is
therefore essential to accurately predict the constrained motions. Many efforts have
Figure 1.10: The First Generation of SeaWEED
Figure 1.11: The Second Generation of SeaWEED
used arbitrary modal shape functions to describe the body deformation, and solved the motion of a hinged system by using the extended motion modes, i.e., additional vertical motion modes perpendicular to the undisturbed free surface were employed to describe motions of hinge joints. The mode expansion technique was adopted by Lee and Newman (2000) to assess the hydroelastic effects on large arrays of hinged structures. Newton-Euler equations of motion with eliminated constraint forces have been utilized to model a multi-body interconnected WEC system by ´ O’Cath´ain et al. (2008) in time domain. The reduced-coordinate approach removes the redundant degrees of freedom due to constraints.
The Lagrange multiplier method, which has been extensively used in the robotics
and gaming industries, is another one to solve constrained motions. Baraff (1996)
presented a Lagrange multiplier based method to simulate constraint motions in the time domain, where various constraints were described by constraint matrices and incorporated into the equations of motions. Constraint motions and forces were solved simultaneously. Catto (2009) used the sequential impulse method to solve multi- constraint motions, where velocities of bodies were first solved without considering the constraints, and the constraint forces were then computed based on the intermediate velocities and constraint matrices to satisfy the constraint conditions.
In ocean engineering field, Sun et al. (2011) applied the Lagrange multiplier technique in the frequency domain. Under linear assumption, a position constraint matrix was combined into equations of motion, and the constrained displacements and forces were obtained directly. The method was employed by Sun et al. (2012) for the dynamic analysis of an installation barge interacting with a substructure of large volume, and by Sun et al. (2016) to predict motions and power of a three- floater WEC. Similar to the method proposed by Baraff (1996), Feng and Bai (2017) investigated the hydrodynamic performance of two freely floating and interconnected barges in the time domain. In their work, the constraints were described by a con- straint matrix which was incorporated into equations of motion, and the nonlinearity of the hydrodynamic forces was taken into account.
1.1.4 Optimization of Simulation-based Design
Simulation-based Design (SBD), or the computer experimental design, can deal
with multi-factor and large-domain problems. Because of the deterministic nature
of computer experiments, no replication, no randomization, and no blocking are re-
quired.
As for design optimization, the procedure typically starts with the determination of design responses, variables, associated domains and levels according to the problem of interest. A set of sampling points are then selected in the design domains. Based on the selected sample points, surrogate models can be constructed to describe the relationship between the design variables and the responses. After the surrogates are verified, the optimized responses, such as the maximum or the minimum values, can then be determined.
1.1.4.1 Selecting Sampling Points
Various sampling methods can be employed to select sampling points on the de- sign domain, for example, the full and fractional factorial design method, the Latin Hypercubes sampling method (McKay et al., 2000) and the uniform design method (Fang et al., 2000).
A full factorial design investigates the effect of all the possible combinations of the factors and levels. This leads to a large number of simulations for problems with more factors and levels. Therefore, fractional factorial design is more practical since it requires less test runs.
In order to conduct the same test with less runs than the factorial designs, the
concept of the space-filling design is proposed to distribute sample points uniformly
in the experimental region (Joseph, 2016). The Latin Hypercube sampling method
(LHS) was first proposed by McKay et al. (2000), which was developed based on the
stratified sampling. The layered feature of the LHS method enables a large number
of input variables and test runs. The method is easy to conduct, and has relatively
small variance (Fang et al., 2000). Moreover, the stratified manner gives the Latin Hypercube Design (LHD) a main advantage that if only one or a few input variables are dominating, the projections onto subspaces will distribute distinctly. However, the LHS method confronts several disadvantages, for example uniform distribution of sampling points in the sample space is not guaranteed. Thus, many efficient extended LHS methods have been proposed to improve the method, such as randomized or- thogonal arrays LHD (Tang, 1993) (Owen, 1994), maximin LHD (Morris et al., 1995), orthogonal LHD (Ye, 1998), uniform LHD (Jin et al., 2003), generalized LHD (Dette et al., 2010), and maximum projection LHD (Joseph, 2016).
Another widely utilized sampling method is the Uniform Design (UD) method, which was proposed by Fang et al. (2000). Unlike the randomly uniform feature in the LHS methods, UD is deterministically uniform. The sampling of UD is based on the Good Lattice Point method. For each n-test-run design with s factors and n levels, there is a unique UD table, U
n(n
m), to determine the sample points, where m is the largest factor number that the design table can deal with (s ≤ m). The uniformity of the UD sampling points is measured by the discrepancy of the sampling points.
To date, many other space-filling design sampling methods have been proposed, such as integrated mean squared error design by Sacks et al. (1989), nested and sliced space-filling design by Qian et al. (2009), fast flexible filling design by Lekivetz et al.
(2015), and bridge design by Jones et al. (2015).
1.1.4.2 Constructing and Exploiting a Surrogate
Based on the selected sample points, surrogate models can be constructed to describe the relationship between the design variables and the responses. Several surrogate modeling methods have been proposed. Examples include the Response Surface method (Box et al., 1951), the Kriging method (Sacks et al., 1989) and the Neural Network Model method (Grossberg, 1988). After the surrogates are verified, the optimized responses, such as the maximum or the minimum values, can then be determined. For unimodal-function surrogate models, a local searcher can be applied.
On the other hand, for multimodal-function surrogate models, a global searcher can be utilized.
Response Surface Methodology (RSM) was proposed by Box and Wilson (1951), which is based on the polynomial model. RSM is generally utilized to analyze the influence of single or multiple input variables to one or several output variables. This method is a sequential procedure that utilizes small steps to find the optimum re- sponses. The basic search procedure of RSM is a local search procedure by using the steepest ascent and steepest descent methods. However, the number of the required sampling data may increase dramatically as the number of the input variables in- crease. To overcome the drawbacks, Derringer et al. (1980) purposed a desirability optimization method which combines the desirability function with RSM to optimize single and multiple responses.
The Kriging method, also called the Gaussian process regression method is also
widely used in simulation-based experiments and spatial analysis. Comparing to the
traditional polynomial model that uses local searchers, the Kriging model is more
suitable for searching in a larger domain utilizing global searchers. The method was first purposed by Krige (1951) and was formally developed by Matheron to conduct spatial analysis (1963). This method utilizes an exact Gaussian process interpolation technique to predict the output variables by calculating the weighted value around the point. In 1989, the Kriging modeling was first introduced to the modeling and optimization in computer experiments (Sacks et al.).
Recently, many new methods are developed based on the Kriging method. For example, the Least Improvement Function method, developed by Sun et al. (2017), applied to structural reliability analysis, and the blind Kriging, developed by Joseph et al. (2008), which is based on a Bayesian variable selection technique, and has a robust performance when dealing with the mis-specification problem (Joseph et al., 2008).
The Neural Network model is generated by adjusting the connection weights be- tween components based on a network function (Grossberg, 1988). The Multilayer Neural Network model (K˚ urkov´a, 1992) and the Radial Basis Function Network model (Chen et al., 1991) are the most widely utilized in Neural Network models. Other types of models include the Multivariate Adaptive Regression Splines (Friedman, 1991), the Least Interpolating Polynomials (De Boor et al., 1990), the Inductive Learning (Langley et al., 1995), the Support Vector Regression (Clarke et al., 2005), and the Multi-point Approximation (Toropov et al., 1993).
The local models are usually utilized to search the optimized responses in small
experimental domain to fit unimodal functions. Examples are the Newton method
(Fischer, 1992) which is an iterative method to find the stationary points of the sur-
(Dennis et al., 1977) which is a modification of Newton method by avoiding comput- ing the Hessian matrix in higher dimensions, simplex algorithm (Dantzig, 2016) for linear optimization, Nelder-Mead method (Olsson et al., 1975) for non-linear opti- mization, and pattern search (derivative-free search) (Hooke et al., 1961) which can be used to search in noncontinuous and non-differentiable spaces.
As for the global searchers, various methods are proposed to search the optima of a non-linear and complicated surrogate. The genetic algorithm was proposed by Holland (1992), which is based on Darwin’s theory of evolution. It is a population- based model that utilizes selection and recombination operators to generate accurate solutions in searching the optima (Whitley, 1994). Similar to genetic algorithm, the particle swarm optimization (Eberhart et al., 1995) also starts with a population of random solutions (particles), and the particles flow through the search domain with randomized velocities, which are determined by their own best position and the overall best position of the entire swarm. The velocities of each particle are kept computed and tuned in each time step until the optimal solution is searched.
1.1.4.3 Wave Energy Converter Optimization
In terms of WEC optimization, the primary objective is to maximize average power extraction for intended operation sites (Khaleghi et al., 2015, Goggins et al., 2014 and Babarit et al., 2005). Basically, geometrical parameters and PTO systems need to be optimized, and several constraints should be set, such as slamming due to large response, the limitation of the WEC motions, and the capacity of the WEC devices (Goggins et al., 2014).
Kofoed et al. (2006) optimized the overall structural geometry, turbines and
reservoir of the Wave Dragon to maximize power production. Babarit et al. (2005) conducted optimization studies on body shape and mechanical features of an oscillat- ing water column WEC, SEAREV, where several constraints were considered, such as static stability, realistic inner pendulum density, and draft of the device.
More recently, Goggins et al. (2014) optimized the geometric shape and the struc- tural radius of an oscillating-body type WEC related to the dynamic heave velocity response of the device. Dai et al. (2017) assessed the performance of an oscillating- body type WEC and optimized the geometry and mechanical parameters of the de- vice. Primary optimization of SeaWEED was conducted by Li et al. (2016), where hinged motions were computed using WAMIT based on the mode expansion method (Newman, 1994). However, since the hinged motions of SeaWEED were described by the vertical movement of joints, it is difficult to incorporate PTO damping into the simulations with WAMIT.
1.2 Overview
In this thesis, a hinged-type wave energy converter, SeaWEED, is introduced.
Potential-flow based time- and frequency-domain programs with the Lagrange multi- plier approach have been developed to simulate the dynamics of constrained multiple bodies, and the numerical results have been validated using model test data. Op- timization studies have been further conducted utilizing the frequency-domain pro- gram by considering various parameters, including damping coefficients of the PTO systems, lengths of truss structures and the draft of the device.
Chapter 2 describes the design of SeaWEED and model tests. Model tests on the
first generation device (1:16 scale) are then briefly introduced. Following, model tests on the second generation device (1:35 scale) are described in detail, including test preparation, instrument calibration, measurement, data-processing, etc.
Chapter 3 derives the mathematical formulations for the constrained multi-body hydrodynamic interactions in the time domain. The Lagrange multiplier approach is utilized to model the constrained dynamics. Nonlinear Froude-Krylov forces are cal- culated over the instantaneous wetted surfaces of the bodies. The numerical method is validated using the test data.
Chapter 4 presents the numerical method for solving the constrained multi-body hydrodynamic interactions in the frequency domain. The numerical results are com- pared against the experimental data.
Chapter 5 elaborates the optimization studies of SeaWEED. Based on the uniform design method for the selection of sample points and the response surface method for surrogate modeling, optimization studies were carried out by considering a variety of parameters, including damping coefficients at the two PTOs, truss lengths, and draft of the device, as independent variables. An optimum combination of these parameters was determined for an intended operation site.
Chapter 6 concludes the current work and brings forward the future work.
Chapter 2
Introduction and Model Tests of SeaWEED
2.1 Introduction of SeaWEED
SeaWEED consists of floating sections linked by trusses or rods and universal joints. The device has semi-submerged floats on the surface of water and inherently faces into the direction of waves. The wave-induced motions of the floats can be converted to electricity through the hydraulic power-take-off (PTO) systems. Floats are connected by trusses which can be customized in length to archive high efficiency for a specified site. The device is considered as an improved attenuator in comparison with Pelamis (Pizer et al., 2000). The design also attempted to address the wave topping and slamming issues encountered by devices such as Pelamis. A complete device has two PTOs and each PTO is located in the back end of a producing module.
Each PTO consists of four double acting hydraulic rams to capture energy from pitch
motions. The use of the rigid truss structure would allow for a higher power output per unit mass, and also reduce the side loading due to tidal currents, local wind or bi-modal swells in comparison with other attenuator devices.
The design of SeaWEED would permit heave, pitch, roll and yaw motions. The electrical power will be only converted from the pitch motion. Inside each module, there is a removable pin. When the pin is removed, each module is allowed to roll independently, which would reduce stress on the entire system thus the risk of failure.
Located in each module, surrounding the mechanical space are positive air-pressure ballast tanks. In the occurrence of extreme sea conditions that could damage the system, air valves on these ballast tanks would open and release air, allowing the tanks to be filled with sea water. Once the sea conditions are improved, the tanks can be re-pressurized with air and the water will be discharged. The system can then return to the normal operating condition.
Furthermore, the device can be customized to operate in various locations by changing its truss lengths and draft. It can utilize on-board electrical generator and standard subsea electrical cables to generate and transmit the energy to shore. In term of applications, an offshore farm of SeaWEED WECs can be utilized to provide electricity to coastal communities and by oil and gas companies to power subsea infrastructure.
The proposed SeaWEED PTOs, currently based on the use of a conventional
hydraulic system, with the utilization of a linear generator being an alternative pos-
sibility, are entirely housed inside the two energy producing modules for maximum
protection. The PTO systems, utilizing a water tight multi-axis joint in the stern
of each module, are driven by movements between interconnected modules, i.e., the
motion between the interconnected modules drives a swashplate located internally (Fig. 2.1) to tilt in various planes and thus to actuate the hydraulic rams. This PTO design is intended not only to protect all PTO components from the harsh marine environment but also to protect the environment from contamination by possible hy- draulic leaks. As shown in Fig. 2.1, each PTO system consists of four hydraulic power capture rams and two horizontal struts. The horizontal struts are primarily for the control of yaw motions but not for energy capturing.
Figure 2.1: SeaWEED PTO System
As for the operational limits, they include both the maximum sea state that a
device is capable of producing electricity in and its method and ability to survive
severe sea conditions. The present design considers methods of decreasing end-stop
issues, through increased flexibility and system damping to increase the operational
limits of the system. This is due in part to being an attenuator but more importantly
is related to how each module in the system connects and flexes. Based on the PTO
design, each section can articulate up to 30 degrees from the neutral axis in any
direction before the end-stop becomes an issue. The proposed PTO design aims to
enable the device to arch over entire wavelengths in high sea states. As indicated above, the purpose of the ballast systems is to allow the SeaWEED to semi-submerge when it is in severe sea conditions. The internal tanks are shown in the illustration and highlighted in green (Fig. 2.2).
Figure 2.2: SeaWEED Ballast Tank
2.2 Model Tests on the First Generation of Sea- WEED
Model tests (scale 1 : 16) of the first generation SeaWEED were conducted by Grey Island Energy Inc. (GIE) at the ocean engineering basin of National Research Council-Institute of Ocean Technology (NRC-IOT) in St. John’s, NL, Canada. The basin (see Fig. 2.3) is 75 m long, 32 m wide and 3.2 m deep. Fig. 2.4 shows the SeaWEED model in the basin.
As shown in Fig. 2.5, the first generation SeaWEED system consists of four equal-
length modules with cambered surfaces connected by three tie-rods with stiffeners.
Figure 2.3: General Arrangement of the Basin (NRC)
Figure 2.4: SeaWEED Model in the NRC-IOT Basin
Table 2.1: Particulars of SeaWEED (first generation)
Parameter Full Scale 1:16 Scale Total Length, L(m) 145.000 9.063 Length of Module, L
m(m) 16.000 1.000 Length of Tie-rod, L
t(m) 27.000 1.688
Width, B(m) 8.800 0.550
Height, H(m) 6.880 0.430
Planned Draft, T
plan(m), 2.500 0.156 Test Draft, T
test(m), 4.480 0.280
The principle dimensions of the first generation SeaWEED and the 1 : 16 model are defined in Fig. 2.5 and Table 2.1. During the tests, the relative pitch angles at PTO-1 were measured. Two hydraulic PTO systems were built to simulate the power capturing procedure of SeaWEED, and the hydraulic damping coefficients were obtained from the flow meter and pressure sensor installed in PTO-1.
Figure 2.5: Body and Tie-rod Length Definition
In the model tests, regular wave periods were 6 s, 8 s, 10 s, and 12 s (full scale).
2.3 Model Tests on the Second Generation of Sea- WEED
The experimental and numerical studies of the first generation model led to the
second generation with improvements in the hull geometry, a lower draft and a dif-
ferent connection structure, which is shown in Fig. 2.7. Model tests on the second
generation of SeaWEED were conducted at the towing tank of Memorial University
(MUN). The towing tank is 58.0 m in length, 4.6 m in width, and 1.8m in depth, as
shown in Fig. 2.6. Fig. 2.7 presents the SeaWEED model in the towing tank. The
1:16 model tests for the first generation of SeaWEED was done by Grey Island Energy
Inc in a large wave tank, and for the second generation of SeaWEED, the model test
scale, 1 : 35, was then determined according to the dimension of the towing tank at
MUN. The principle dimensions of SeaWEED and the 1 : 35 model are listed in Table
2.2.
Figure 2.6: Towing Tank of MUN
Figure 2.7: SeaWEED Model in the Towing Tank
Table 2.2: Particulars of SeaWEED (second generation)
Parameter Full Scale 1:35 Scale Length of Nose Module, L
n(m) 9.000 0.257
Length of Tail Module, L
t(m) 9.000 0.257 Length of Producing Module, L
p(m) 16.000 0.457
Width, B(m) 8.000 0.229
Height, H(m) 5.000 0.143
Draft, T (m), 2.50 0.0714
2.3.1 Test Matrix
In the model tests, the wave periods were varied from 5.5 s to 10.0 s (full scale), and the wave steepness was set as 1/50. The definition of the body length is shown in Fig. 2.8. The combinations of truss lengths are presented Table 2.3, and the draft was 2.5 m in full scale. The test matrix is presented in Table 2.4. The corresponding PTO damping settings are given in Table 2.5. Several repeated tests were carried out at a few wave frequencies around the region with maximum responses. Good repeatability was achieved in tests.
Figure 2.8: Body and Truss Length Definition
Table 2.3: Truss length combinations
Case
Full Scale 1:35 Scale
L
1(m) L
2(m) L
3(m) L
total(m) L
1(m) L
2(m) L
3(m) L
total(m) T 1 46.912 43.912 43.912 134.736 1.340 1.255 1.255 3.850 T 2 41.252 43.912 49.962 135.126 1.179 1.255 1.427 3.861 T 3 41.252 43.912 56.402 141.566 1.179 1.255 1.611 4.045 T 4 41.252 43.912 63.233 148.397 1.179 1.255 1.807 4.240
Table 2.5: Damping cases
Spring Compression (mm)
Case No. PTO-1 PTO-2
D0 - -
D1 3.0 2.3
D2 4.3 4.0
D3 4.3 -
D4 - 4.0
D5 6.0 5.0
Table 2.4: Test matrix
No.
T (s) No. of Conducted Tests
Full Scale
Model Scale
T 1 D0
T 2 D0
T 3 D0
T 4 D0
T 1 D1
T 1 D2
T 1 D3
T 1 D4
T 1 D5
1 5.50 0.93 1 1 1 1 0 0 0 0 0
2 6.00 1.01 1 1 1 3 0 0 0 0 0
3 6.50 1.10 1 1 1 3 0 0 0 0 0
4 7.00 1.18 3 1 3 1 0 0 0 0 0
5 7.50 1.27 3 3 3 3 3 3 3 3 3
6 8.00 1.35 3 3 3 3 2 3 1 2 3
7 8.50 1.44 3 3 3 1 1 1 3 3 1
8 9.00 1.52 3 1 1 3 1 1 3 3 1
9 10.00 1.69 1 3 1 1 1 1 1 1 1
T 1 − T 4: Body length combinations; D0 − D5: PTO Damping cases
2.3.2 Test Set-up
The test set-up is given in Fig. 2.9. The model was constrained by four soft mooring lines from drift motions. Four AWP-24 resistive wave probes were used to measure the wave elevations. Motions of the model were measured by a Qualisys motion capture system. Fig. 2.7 shows the scaled model with tracking markers distributed on its floats and trusses.
Figure 2.9: Test Set-up
The PTO systems were mimicked by friction dampers. As shown in Fig. 2.10, the angular hinge motions of the model were transferred to the linear motion of the sliding bar of the damper by the swash plate. Honeywell Model 31 load cells and the Contelec linear position sensors were used to measure the friction forces and the sliding motions, respectively.
As shown in Fig. 2.11, the damper is consisted of a steel frame, an aluminum
Figure 2.10: Friction Damper
sliding bar, a spring, and two Teflon bars. The aluminum bar is guided by four low-friction bearings. The upper Teflon bar is pressed by the spring, which can be adjusted using the screw on top of the frame to change the friction forces. To quantify the adjustments, marks were also machined to indicate the spring displacement. It is noted that friction dampers with aluminum-leather and aluminum-steel contacts were also tested. However, the former suffered from inconsistent frictions due to the change of humidity, and the latter failed to provide smooth frictions since the contact surfaces were not perfectly even. The damper with the Teflon-aluminum contact showed good repeatability. Two identical dampers were manufactured to mimic the two PTO systems.
2.3.3 Pre-tests
Before the model tests, the instruments including wave probes, load cells and
displacement sensors were calibrated. The Qualisys system was calibrated according
to the calibration quality indicator in the Qualisys Track Manager software. For each
damping set-up, the friction dampers were calibrated before being installed on the
Figure 2.11: Friction Damper
model.
2.3.3.1 Wave Probe Calibration
Static calibrations were preformed on the AWP-24 wave probes. The voltages were measured when the wave probes were submerged in five different depths, as shown in Fig. 2.12. For each wave probe, the calibration was repeated for three times. An example of the calibration results for WP3 is presented in Fig. 2.13.
2.3.3.2 Load Cell Calibration
Two Honeywell Model-31 load cells (see Fig. 2.14) were used to measure the forces
in the PTO system. Three static calibrations were repeated for each load cell. In
each calibration, six load steps were used in compressing and expansion directions,
and each step was kept stable for 30 seconds. The calibration results for the load cell
in PTO-2 are presented in Fig. 2.15.
Figure 2.12: Wave Probe Calibration
0 100 200 300 400 500 600
−4 −3 −2 −1 0 1 2 3 4
Depth (mm)
Voltage (V)
Mean Calibration Curve Calibration 1 Calibration 2 Calibration 3
Figure 2.13: Wave Probe Calibration Results (WP3)
Figure 2.14: Honeywell Model-31 Load Cell
−20
−15
−10
−5 0 5 10 15 20
−0.03 −0.02 −0.01 0 0.01 0.02 0.03
Weight (kg)
Voltage (V)
Mean Calibration Curve Calibration 1 Calibration 2 Calibration 3
Figure 2.15: Load Cell Calibration Results (PTO-2)
2.3.3.3 Displacement Sensor Calibration
Two Contelec linear position sensors with ranges of 50 mm and 75 mm were used to measure the damper sliding bar movements. The sensors were calibrated by recording the output voltage when different displacements were applied. Each displacement sensor was calibrated for three time. Fig. 2.16 shows the calibration results of the displacement sensor in PTO-2.
0 10 20 30 40 50 60 70 80
0 2 4 6 8 10
Displacement (mm)
Voltage (V)
Mean Calibration Curve Calibration 1 Calibration 2 Calibration 3
Figure 2.16: Displacement Sensor Calibration Results (PTO-2)
2.3.3.4 Damper Calibration
As shown in Fig. 2.17, a calibration frame was designed and built to calibrate the
dampers before the model tests. The motor was turned to a fixed revolving speed by
a controller. The speed was recorded by an RPM sensor. A 1:20 gearbox converted
the speed to the disk system speed, and a slider-crank mechanism transformed the rotational motion to the translational motion of the sliding bar. The amplitude of the bar movement was controlled by changing the rotating diameter of the stick connecting to the disk. The displacements and the friction forces were measured by a displacement sensor paralleled to the bar and a load cell on the bar.
Figure 2.17: Damper Calibration Apparatus
Three damper set-ups were conducted, and the springs of two dampers were ad-
justed to a certain compression to achieve the target damping forces. The damping
coefficients from the calibration and the tests of two damping systems are presented with respect to the velocity amplitude of the sliding motion in Fig. 2.18, Fig. 2.19, and Fig. 2.20.
0 2000 4000 6000 8000 10000 12000
0 0.005 0.01 0.015 0.02 0.025 0.03
Damping Coefficient (Ns/m)
Velocity (m/s)
PTO 1 Calibration PTO 2 Calibration PTO 1 Experimental PTO 2 Experimental
Figure 2.18: Damper Calibration Results (D1)
2.3.3.5 Sychronization of Measurements
During tests, the measurements of wave amplitudes and friction forces and dis-
placements of the dampers are recorded by the LabView system, and the motions
of SeaWEED are measured by the Qualisys system. These measurements need to
be synchronized. An electrical synchronization signal was generated and recorded in
the data collection by the two systems, i.e., the LabView system and the Qualisys
system.
0 5000 10000 15000 20000
0 0.005 0.01 0.015 0.02 0.025 0.03
Damping Coefficient (Ns/m)
Velocity (m/s)
PTO 1 Calibration PTO 2 Calibration PTO 1 Experimental PTO 2 Experimental
Figure 2.19: Damper Calibration Results (D2, D3 and D4)
0 5000 10000 15000 20000 25000
0 0.005 0.01 0.015 0.02 0.025 0.03
Damping Coefficient (Ns/m)
Velocity (m/s)
PTO 1 Calibration PTO 2 Calibration PTO 1 Experimental PTO 2 Experimental
Figure 2.20: Damper Calibration Results (D5)
The Qualisys system can receive a square wave voltage signal as a trigger via the Oqus sync unit of the Qualisys system. The triggering signal was generated by the National Instruments NI-9264 module installed on the data acquisition system. The motion measurements were then recorded by the Qualisys system after a preset delay (20 ms in the present tests). In the present model tests, a square wave signal with an amplitude of 5 V and a duration of 50 ms was generated as shown in Fig. 2.21.
Based on the triggering signal, the delay between the two measurement systems can be precisely determined, and a re-alignment can be performed on all the measurements.
−1 0 1 2 3 4 5 6
0.18 0.185 0.19 0.195 0.2 0.205 0.21
Synchronization Signal (V)
Time (s)
Input Signal Trigger Point
Figure 2.21: Synchronization Input by Module NI9264
2.3.4 Experimental Data-processing
As shown Fig. 2.22, the experimental relative pitch angles, θ
1and θ
2, at the
joints of two PTO systems, P T O − 1 and P T O − 2, were captured and measured
by a Qualisys motion capture system. FFT analysis was conducted to obtain the amplitudes of angular motions.
Figure 2.22: Relative Pitch Angles
When the PTO systems were activated, the damping forces and the displacements of the sliding bars were measured. The velocities of the bars were obtained from time series of their displacements. The moving average method was applied to remove unrealistic oscillations in the time series of force and velocity. Fig. 2.23 presents a segment of time series of force and velocity for PTO-1. The corresponding wave period was 1.35 s in model scale.
60 61 62 63 64 65 66 67 68 69 70
−50 0 50
F (N)
t (s) PTO1 Damping Force
60 61 62 63 64 65 66 67 68 69 70
−0.01 0 0.01
V (m/s)
t (s) PTO1 Sliding Velocity
Figure 2.23: Damping Force and Sliding Velocity
The average power absorption of the ith PTO, denoted as P
i, is P
i=
R
T0
F
i(t)v
i(t)dt
T (2.1)
where F
i(t) is the damping force, v
i(t) is the sliding velocity, and T is the time length.
Then the equivalent damping coefficient, d
i, was obtained as d
i=
R
T0
F
i(t)v
i(t)dt R
T0
v
i2(t)dt (2.2)
Introducing the total power of the incident wave per unit crest length across the device,
E = ρg
2A
24ω (2.3)
the power capture width for the SeaWEED is given as C
width=
P
2 i=1P
iE (2.4)
where A is the wave amplitude and ω is the wave frequency.
In order to map the model test results to the full scale device, the similitude
theory is applied. Two main similarity parameters are involved, namely, the Strouhal
number, S =
U TL, and the Froude number, F r =
√UgL, where L is the length of
the device, U is the wave speed, T is the wave period and g is the gravitational
acceleration. Denote a quantity of interest, such as relative pitch angle, damping
coefficient and capture width, as Q, the relationship between the model scale value
and the full scale one can be expressed by Q
f ull= Q
model∗ λ
n, where λ is the scale
and n is the scaling factor derived based on the similarity parameters. The scaling of
the quantities involved in the present study are listed in Table 2.6.
Table 2.6: Scaling of quantities
Quantity, Q Unit Scaling factor, n
Device dimensions (length, breadth and draft), L, B and T (m) 1.0
Wave period, T (s) 0.5
Damping coefficient, d (Nms) 4.5
Velocity, U (m/s) 0.5
Relative pitch angle, θ − 0.0
Capture width, C
w(m) 1.0
Absorbed power, P
s(kW ) 3.5
Chapter 3
Time-domain Simulation of SeaWEED
The three bodies of SeaWEED articulated by hinge joints are subjected to con-
strained motions. To simulate the dynamics of SeaWEED, a potential-flow based
time-domain program is developed. This chapter presents the mathematical for-
mulations of the time-domain method where the constraints are modeled using the
Lagrange multiplier approach (Baraff,1996), the Froude-Krylov forces are calculated
over the instantaneous wetted surfaces of the bodies and the wave pressure on the
body surfaces is computed applying the Wheeler Stretching method (Wheeler et al.,
1969). The numerical method is validated using the model test data, and good agree-
ment is achieved.
3.1 Mathematical Formulation
3.1.1 Coordinate System Definition
As shown in Fig. 3.1, an earth-fixed Cartesian coordinate system, OXY Z, is employed with the OXY plane coinciding with the undisturbed free surface and the OZ axis pointing vertically upward. In three body-fixed coordinate systems, o
ix
iy
iz
i, i = 1, 2, 3, o
iis at the point of intersection of calm water surface, the longitudinal plane of symmetry, and the vertical plane passing through the centre of gravity (CG) of the i
thbody; the o
iz
iaxis points upward; the o
ix
iy
iplane coincides with the undisturbed free surface when the body is at rest; and the o
ix
iaxis points from the tail module to the nose module.
Figure 3.1: Coordinate Systems
3.1.2 Equations of Motion
According to the work of Danmeier (1999) and Qiu and Peng (2013), equations
of motion for the SeaWEED system can be developed as follows.
Denoting a column vector by braces {}, translational displacements of the ith body in the OXY Z system are represented by X X X
i= {X
i,1, X
i,2, X
i,3} and the Eulerian angles are given by X X X
Ri= {X
i,4, X
i,5, X
i,6}. The angular velocity in o
ix
iy
iz
iis denoted by ω ω ω
i. The time rate change of the Eulerian angles is related to the angular velocity by
X X X ˙ ˙ ˙
Ri= T T T
iω ω ω
i=
1 s
1t
2c
1t
20 c
1−s
10 s
1/c
2c
1/c
2
ω ω ω
i(3.1)
where c
k= cos(X
i,3+k), s
k= sin(X
i,3+k) and t
k= tan(X
i,3+k) for k=1,2 and 3.
Equations of motion for the ith body are then given as
m m
m
i−m
ix x x
cgiD D D
Tim
ix x x
cgiD D D
iIII
oi
X ¨ X X
i˙ ω ω ω
i
=
F F F
iM M M
oi
(3.2)
where m m m
iis the 3 × 3 matrix with the body mass, m
i, along its diagonal and zero everywhere else, III
oiis the mass moment of inertia matrix with respect to the origin of the ith body, x x x
cgiis the centre of gravity of ith body, F F F
iare the external forces acting on the body in OXY Z, M M M
oiare the moment about the origin of the body- fixed coordinate system, and D D D
iis the rotational transformation matrix between the earth-fixed and body-fixed coordinate systems as follows:
D D D
i=
c
2c
3c
2s
3−s
2s
1s
2c
3− c
1s
3s
1s
2s
3+ c
1c
3s
1c
2c
1s
2c
3+ s
1s
3c
1s
2s
3− s
1c
3c
1c
2
(3.3)
Equation 3.2 can also be written in a concise form as below:
where F F F
Eiis the vector including resultant forces and moments on the ith body, and vvv
i= { X ˙
i1, X ˙
i2, X ˙
i3, ω
i1, ω
i2, ω
i3}, i = 1, 2, 3 (3.5) The total force acting on the ith body can be written as
F
F F
Ei= F F F
F Ki+ F F F
RSi+ F F F
Ri+ F F F
Di+ F F F
P T Oi+ F F F
Ci(3.6) where F F F
F Kiare the Froude-Krylov forces; F F F
RSiare the restoring forces; F F F
Riand F F F
Diare the forces due to radiated and diffracted waves, respectively; F F F
P T Oiare the damping forces from the PTO system; and F F F
Ciare the constraint forces due to hinge joints.
The nonlinear Froude-Krylov forces are calculated according to instantaneous wet- ted surface applying the Wheeler Stretching Approach (Wheeler et al., 1969). The lin- ear diffraction forces are obtained from the frequency domain solution using WAMIT.
The linear radiation forces on the ith body are calculated using the impulse function method and the added mass and damping matrices from WAMIT, i.e.,
F
F F
Ri= −A A A(∞)
ijx x x ¨
j(t) − Z
t−∞
K K K
ij(t − τ ) ˙ x x x
j(τ )dt (3.7) where A A A(∞)
ijis the added mass matrix (6 × 6) at the infinite frequency of the ith body due to the jth body, i = 1, 2, 3 and j = 1, 2, 3 for SeaWEED, ˙ x x x
j(τ) and ¨ x x x
j(t) are the velocities and accelerations of the jth body, respectively, K K K
ij(t − τ) is the impulse function of the ith body due to the jth body, which is also a 6 × 6 matrix.
The response function can be obtained from the damping matrix, B B B
ij(ω), which is a function of wave frequency ω and was calculated from WAMIT in this work, i.e.,
K K K
ij(τ ) = 2 π
Z
∞0
